The Askey-Wilson Function is the Laplace Transform of a Positive Measure

Abstract

The Askey-Wilson function $ϕ(x, γ)$ is a meromorphic continuation of the Askey-Wilson polynomials in their spectral variable off the lattice γ= ã q^m, m ∈ ℤ₊. We show that the analytic part of the Askey-Wilson function ϕ^an(x, ã q^γ) is the Laplace transform of a positive measure of total measure ϕ^an(x, s₀) < ∞, under certain conditions on the parameters governing the function. We deduce log-convexity of this function, as well as positive-definiteness and an integral formula for a certain intertwining operator associated with the Askey-Wilson operator L. This offers a q-analogue of several results, in the trigonometric limit due to Sahi-McSwiggen, and in the rational (i.e. Dunkl) limit due to Rösler.

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BibTeX RIS

BibTeX

@misc{agulnick2026askeywilson,
  title = {{The Askey-Wilson Function is the Laplace Transform of a Positive Measure}},
  author = {Aaron Agulnick},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03907},
  eprint = {2611.03907},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03907},
}

RIS

TY  - UNPB
TI  - The Askey-Wilson Function is the Laplace Transform of a Positive Measure
AU  - Aaron Agulnick
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03907
UR  - https://arxiv.org/abs/2611.03907
AB  - The Askey-Wilson function $ϕ(x, γ)$ is a meromorphic continuation of the Askey-Wilson polynomials in their spectral variable off the lattice γ= ã q^m, m ∈ ℤ_+. We show that the analytic part of the Askey-Wilson function ϕ^an(x, ã q^γ) is the Laplace transform of a positive measure of total measure ϕ^an(x, s_0) < ∞, under certain conditions on the parameters governing the function. We deduce log-convexity of this function, as well as positive-definiteness and an integral formula for a certain intertwining operator associated with the Askey-Wilson operator L. This offers a q-analogue of several results, in the trigonometric limit due to Sahi-McSwiggen, and in the rational (i.e. Dunkl) limit due to Rösler.
ER  -

Abstract

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